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Theorem addid1 9202
Description:  0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
addid1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )

Proof of Theorem addid1
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 9046 . 2  |-  1  e.  RR
2 ax-rnegex 9017 . 2  |-  ( 1  e.  RR  ->  E. c  e.  RR  ( 1  +  c )  =  0 )
3 ax-1ne0 9015 . . . . . 6  |-  1  =/=  0
4 oveq2 6048 . . . . . . . . . 10  |-  ( c  =  0  ->  (
1  +  c )  =  ( 1  +  0 ) )
54eqeq1d 2412 . . . . . . . . 9  |-  ( c  =  0  ->  (
( 1  +  c )  =  0  <->  (
1  +  0 )  =  0 ) )
65biimpcd 216 . . . . . . . 8  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
( 1  +  0 )  =  0 ) )
7 oveq2 6048 . . . . . . . . 9  |-  ( ( 1  +  0 )  =  0  ->  (
( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
8 ax-icn 9005 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
98, 8mulcli 9051 . . . . . . . . . . . . . 14  |-  ( _i  x.  _i )  e.  CC
109, 9mulcli 9051 . . . . . . . . . . . . 13  |-  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC
11 ax-1cn 9004 . . . . . . . . . . . . 13  |-  1  e.  CC
12 0cn 9040 . . . . . . . . . . . . 13  |-  0  e.  CC
1310, 11, 12adddii 9056 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
1410mulid1i 9048 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  =  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )
15 mul01 9201 . . . . . . . . . . . . . . 15  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC  ->  ( (
( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0 )
1610, 15ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0
17 ax-i2m1 9014 . . . . . . . . . . . . . 14  |-  ( ( _i  x.  _i )  +  1 )  =  0
1816, 17eqtr4i 2427 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  ( ( _i  x.  _i )  +  1 )
1914, 18oveq12i 6052 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2013, 19eqtri 2424 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2120, 16eqeq12i 2417 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )  =  0 )
2210, 9, 11addassi 9054 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )
239mulid1i 9048 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  1 )  =  ( _i  x.  _i )
2423oveq2i 6051 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )
259, 9, 11adddii 9056 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1 ) )
2617oveq2i 6051 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( _i  x.  _i )  x.  0
)
27 mul01 9201 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  x.  _i )  e.  CC  ->  (
( _i  x.  _i )  x.  0 )  =  0 )
289, 27ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  0 )  =  0
2926, 28eqtri 2424 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  0
3025, 29eqtr3i 2426 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  0
3124, 30eqtr3i 2426 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( _i  x.  _i ) )  =  0
3231oveq1i 6050 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( 0  +  1 )
3322, 32eqtr3i 2426 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1
) )  =  ( 0  +  1 )
34 00id 9197 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
3534eqcomi 2408 . . . . . . . . . . 11  |-  0  =  ( 0  +  0 )
3633, 35eqeq12i 2417 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )  =  0  <->  ( 0  +  1 )  =  ( 0  +  0 ) )
37 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
38 readdcan 9196 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
391, 37, 37, 38mp3an 1279 . . . . . . . . . 10  |-  ( ( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 )
4021, 36, 393bitri 263 . . . . . . . . 9  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  1  =  0 )
417, 40sylib 189 . . . . . . . 8  |-  ( ( 1  +  0 )  =  0  ->  1  =  0 )
426, 41syl6 31 . . . . . . 7  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
1  =  0 ) )
4342necon3d 2605 . . . . . 6  |-  ( ( 1  +  c )  =  0  ->  (
1  =/=  0  -> 
c  =/=  0 ) )
443, 43mpi 17 . . . . 5  |-  ( ( 1  +  c )  =  0  ->  c  =/=  0 )
45 ax-rrecex 9018 . . . . 5  |-  ( ( c  e.  RR  /\  c  =/=  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
4644, 45sylan2 461 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
47 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  A  e.  CC )
48 simplrl 737 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  RR )
4948recnd 9070 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  CC )
5047, 49mulcld 9064 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  x.  x )  e.  CC )
51 simplll 735 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  RR )
5251recnd 9070 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  CC )
5312a1i 11 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  CC )
5450, 52, 53adddid 9068 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( ( A  x.  x )  x.  c )  +  ( ( A  x.  x )  x.  0 ) ) )
5511a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  CC )
5655, 52, 53addassd 9066 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 1  +  ( c  +  0 ) ) )
57 simpllr 736 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  c )  =  0 )
5857oveq1d 6055 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 0  +  0 ) )
5956, 58eqtr3d 2438 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 0  +  0 ) )
6034, 59, 573eqtr4a 2462 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 1  +  c ) )
6137a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  RR )
6251, 61readdcld 9071 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  e.  RR )
631a1i 11 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  RR )
64 readdcan 9196 . . . . . . . . . . 11  |-  ( ( ( c  +  0 )  e.  RR  /\  c  e.  RR  /\  1  e.  RR )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6562, 51, 63, 64syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6660, 65mpbid 202 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  =  c )
6766oveq2d 6056 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( A  x.  x )  x.  c ) )
6854, 67eqtr3d 2438 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( ( A  x.  x )  x.  c ) )
69 mul31 9190 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC  /\  c  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
7047, 49, 52, 69syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
71 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  x.  x )  =  1 )
7271oveq1d 6055 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( c  x.  x
)  x.  A )  =  ( 1  x.  A ) )
7347mulid2d 9062 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  x.  A )  =  A )
7470, 72, 733eqtrd 2440 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  A )
75 mul01 9201 . . . . . . . . 9  |-  ( ( A  x.  x )  e.  CC  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7650, 75syl 16 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7774, 76oveq12d 6058 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( A  + 
0 ) )
7868, 77, 743eqtr3d 2444 . . . . . 6  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  +  0 )  =  A )
7978exp42 595 . . . . 5  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( x  e.  RR  ->  ( (
c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) ) )
8079rexlimdv 2789 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( E. x  e.  RR  ( c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) )
8146, 80mpd 15 . . 3  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
8281rexlimiva 2785 . 2  |-  ( E. c  e.  RR  (
1  +  c )  =  0  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
831, 2, 82mp2b 10 1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947   _ici 8948    + caddc 8949    x. cmul 8951
This theorem is referenced by:  cnegex  9203  addid2  9205  addcan2  9207  addid1i  9209  addid1d  9222  subid  9277  subid1  9278  shftval3  11846  reim0  11878  isercolllem3  12415  fsumcvg  12461  summolem2a  12464  ovolicc1  19365  relexpadd  25091  risefac1  25299  brbtwn2  25748  axsegconlem1  25760  ax5seglem4  25775  axeuclid  25806  axcontlem2  25808  axcontlem4  25810  stoweidlem26  27642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081
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